Multiply polynomial $ \, \color{blue}{ x^5+x^2+x+1 } \, $ by $ \, \color{orangered}{ x^3+1 }$.

$$ \left( \color{blue}{x^5+x^2+x+1} \right) \cdot \left( \color{orangered}{x^3+1} \right) = x^8+2x^5+x^4+x^3+x^2+x+1 $$

We can multiply polynomials by using a GRID METHOD

Write one of the polynomials across the top and the other down the left side.

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{x^5} & \color{blue}{x^2} & \color{blue}{x} & \color{blue}{1} \\ \hline \color{blue}{x^3} & & & & \\ \hline \color{blue}{1} & & & & \\ \hline \end{darray} $$

Fill in each empty box by multiplying the intersecting terms.

$$ \begin{darray}{|c|c|c|c|c|}\hline & \color{blue}{x^5} & \color{blue}{x^2} & \color{blue}{x} & \color{blue}{1} \\ \hline \color{blue}{x^3} & \color{orangered}{x^8} & \color{orangered}{x^5} & \color{orangered}{x^4} & \color{orangered}{x^3} \\ \hline \color{blue}{1} & \color{orangered}{x^5} & \color{orangered}{x^2} & \color{orangered}{x} & \color{orangered}{1} \\ \hline \end{darray} $$

Combine like terms:

$$ x^8 + x^5 + x^5 + x^4 + x^2 + x^3 + x + 1 = \\ x^8+2x^5+x^4+x^3+x^2+x+1 $$

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